Theorem preimafvsspwdm 43598

Description: The class 𝑃 of all preimages of function values is a subset of the power set of the domain of the function. (Contributed by AV, 5-Mar-2024.)

Hypothesis

Ref	Expression
setpreimafvex.p	⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}

Assertion

Ref	Expression
preimafvsspwdm	⊢ (𝐹 Fn 𝐴 → 𝑃 ⊆ 𝒫 𝐴)

Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝑥,𝑃

Allowed substitution hint:   𝑃(𝑧)

Proof of Theorem preimafvsspwdm

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		setpreimafvex.p	. . . 4 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
2	1	elsetpreimafvssdm 43595	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑠 ∈ 𝑃) → 𝑠 ⊆ 𝐴)
3	2	ralrimiva 3182	. 2 ⊢ (𝐹 Fn 𝐴 → ∀𝑠 ∈ 𝑃 𝑠 ⊆ 𝐴)
4		pwssb 5023	. 2 ⊢ (𝑃 ⊆ 𝒫 𝐴 ↔ ∀𝑠 ∈ 𝑃 𝑠 ⊆ 𝐴)
5	3, 4	sylibr 236	1 ⊢ (𝐹 Fn 𝐴 → 𝑃 ⊆ 𝒫 𝐴)

Syntax hints:   → wi 4   = wceq 1537  {cab 2799  ∀wral 3138  ∃wrex 3139   ⊆ wss 3936  𝒫 cpw 4539  {csn 4567  ^◡ccnv 5554   “ cima 5558   Fn wfn 6350  ‘cfv 6355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-xp 5561  df-cnv 5563  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-fn 6358

This theorem is referenced by: (None)